In this problem we want to evaluate the partial derivative of Gibb's Free Energy with respect
to pressure.
And we want to do this partial derivative with respect to temperature and we want to
end up with something that is in terms of variables we can measure.
So we will start with the definition of Gibb's Free Energy.
Gibb's Free Energy just H - TS.
And now if we take and just directly apply it to this, the partial G with respect to
pressure at constant Temperature becomes the partial of H with respect to pressure at constant
T. So since temperature constant we can write it, constant S with respect to P at constant
T. So notice we just have T pulled out of the derivative because it was a constant for
this calculation.
Now we want to simplify this to write it in terms of other variables, because we want
it in terms of measurable variables.
So we are going to use one of the Maxwell relations.
So this is the partial of S with respect to Pressure at constant T is equal to partial
of volume with respect to temperature at constant pressure.
So this is one of the standard Maxwell Relationships.
The other we have to determine, again, something that was derived elsewhere, and we can talk
about where this came from, but let's write it down first.
So you can see we now have these two terms, these are in terms of measurable variables,
so we are essentially finished if we can determine where this second equation came from.
So let's go back and look at the differential expression for the first law, for a closed
system.
And if we do this reversibly, dQ is TdS and dW is PdV.
And now since H is U + PV, dH is dU + PdV + VdP.
So dH is dU is here, TdS - PdV+ PdV +VdP.
So these cancel, so we have the differential of H in terms of entropy and pressure.
So we are partway there.
We want the partial of H with respect to P at constant T. So what we are going to do
is apply the expansion rule to this.
And the expansion rule says I can take this derivative, differential, and I can write
is like so.
So.
This sort of looks like we divided both sides by dP, made them all partial derivatives and
then T constant.
It's actually more complicated than that but that is certainly an easy way to remember
that and you can check a textbook for more detailed explanation.
But the expansion rule gives us this which we can simplify T dSdP constant T +V *1.
Partial P with respect to P is 1.
And now notice this partial of S with respect to P we had this Maxwell Relation so we can
substitute this Maxwell relation here.
Now I've made the substitution, now I have the partial of H with respect ot P.
This is the equation I wrote down up here, so essentially now I can substitute this equation
and this equation into this equation and I'll be finished.
So I have just made the substitution of those two equations into our original equation and
then we can see this term and this term are going to cancel.
And we will end up with the Gibb's Free Energy changes with pressure and constant temperature
is equal to the Volume.
And with many of these calculations, knowing how to get to the answer already makes it
much easier to do to the calculation because we have to go in and use Maxwell Relations,
have to go back and go to some of our original definitions, and then here, using the expansion
rule,so they are not always obvious, the approach to get to our final answer.